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Pattern Formation

  • Till Frank
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

This chapter consists of two parts. In the first part, the fundamental mechanism of pattern formation will be explained. In this context, the notion of a pattern will be defined explicitly. In fact, it will be shown that pattern formation involves two different but related types of patterns: attractor and repellor patterns, on the one hand, and basis patterns, on the other. Attractor and repellor patterns can come, for example, as fixed point patterns. In this case, we are dealing with fixed point attractor and fixed point repellor patterns. Attractor patterns correspond to the patterns that we humans form with our bodies (e.g., when walking) or to brain activity patterns in the human brain. Attractor patterns corresponds to the patterns that are usually observed in experiments in the first place. Attractor patterns can turn into repellor patterns at bifurcation points, and in this context are like two sides of the same coin. Attractor patterns are composed of elementary units: the basis patterns. In special cases, an attractor pattern may be composed of a single basis pattern. In such cases, an attractor pattern is identical to a basis pattern. In the context of the basis patterns the abstract concept of eigenvalues will be introduced. By definition, these eigenvalues determine how quickly basis patterns emerge and disappear. However, taking all eigenvalues together as a set, the set of eigenvalues constitutes an eigenvalue spectrum. This spectrum is the key for understanding pattern formation in general, and transitions between attractor patterns, the emergence of attractor patterns, and bifurcations, in particular. That is, eigenvalues play a key role in the theory of pattern formation and synergetics. Finally, in the context of basis patterns also the concept of pattern amplitudes will be introduced. As will be explained below, pattern amplitudes describe how much basis patterns contribute to attractor patterns. Pattern amplitudes, in general, and the so-called reduced amplitude space, in particular, allow for a convenient description of the formation of patterns and transitions between attractor patterns. The second part of this chapter is devoted to the Lotka-Volterra-Haken amplitude equations. These equations describe pattern formation in the aforementioned reduced amplitude space. The Lotka-Volterra-Haken amplitude equations correspond to a general class of amplitude equations in the theory of pattern formation. They will be used in all applications of this book to describe the formation of brain and body activity patterns in humans and animals (i.e., “perception”, “cognition”, and “behavior”).

References

  1. 20.
    M. Begon, J.L. Harper, C.R. Townsend, Ecology, Individuals, Populations and Communities (Blackwell Scientific Publications, Boston, 1990)Google Scholar
  2. 22.
    M. Bestehorn, R. Friedrich, Rotationally invariant order parameter equations for natural patterns in nonequilibrium systems. Phys. Rev. E 59, 2642–2652 (1999)ADSMathSciNetCrossRefGoogle Scholar
  3. 23.
    M. Bestehorn, H. Haken, Associative memory of a dynamical system: an example of the convection instability. Z. Phys. B 82, 305–308 (1991)ADSCrossRefGoogle Scholar
  4. 24.
    M. Bestehorn, R. Friedrich, H. Haken, Two-dimensional traveling wave patterns in nonequilibrium systems. Z. Phys. B 75, 265–274 (1989)ADSCrossRefGoogle Scholar
  5. 30.
    L.L. Bonilla, C.J. Perez-Vicente, R. Spigler, Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions. Phys. D 113, 79–97 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 36.
    P.C. Bressloff, Neural networks, lattice instantons, and the anti-integrable limit. Phys. Rev. Lett. 75, 962–965 (1995)ADSCrossRefGoogle Scholar
  7. 37.
    P.C. Bressloff, P. Roper, Stochastic dynamics of the diffusive Haken model with subthreshold periodic forcing. Phys. Rev. E 58, 2282–2287 (1998)ADSCrossRefGoogle Scholar
  8. 57.
    M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)ADSzbMATHCrossRefGoogle Scholar
  9. 58.
    G.C. Cruywagen, P.K. Maini, J.D. Murray, Biological pattern formation on two-dimensional spatial domains: a nonlinear bifurcation analysis. SIAM J. Appl. Math. 57, 1485–1509 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 61.
    A. Daffertshofer, H. Haken, A new approach to recognition of deformed patterns. Pattern Recogn. 27, 1697–1705 (1994)CrossRefGoogle Scholar
  11. 64.
    F.J. Diedrich, W.H. Warren, Why change gaits? Dynamics of the walk-run transition. J. Exp. Psychol. - Hum. Percept. Perform. 21, 183–202 (1995)CrossRefGoogle Scholar
  12. 74.
    V. Dufiet, J. Boissonade, Dynamics of Turing pattern monolayers close to onset. Phys. Rev. E 53, 4883–4892 (1996)ADSCrossRefGoogle Scholar
  13. 76.
    A.K. Dutt, Turing pattern amplitude equations for a model glycolytic reaction-diffusion system. J. Mater. Chem. 48, 841–855 (2010)MathSciNetzbMATHGoogle Scholar
  14. 78.
    R.E. Ecke, H. Haucke, Y. Maeno, J.C. Wheatley, Critical dynamics at a Hopf bifurcation to oscillatory Rayleigh-Benard convection. Phys. Rev. A 33, 1870–1878 (1986)ADSCrossRefGoogle Scholar
  15. 82.
    B. Fiedler, T. Gedeon, A class of convergent neural network dynamics. Phys. D 111, 288–294 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 90.
    T.D. Frank, On a multistable competitive network model in the case of an inhomogeneous growth rate spectrum with an application to priming. Phys. Lett. A 373, 4127–4133 (2009)ADSzbMATHCrossRefGoogle Scholar
  17. 95.
    T.D. Frank, New perspectives on pattern recognition algorithm based on Haken’s synergetic computer network, in Perspective on Pattern Recognition ed. by M.D. Fournier, pp. 153–172, Chap. 7 (Nova Publ., New York, 2011)Google Scholar
  18. 96.
    T.D. Frank, Rate of entropy production as a physical selection principle for mode-mode transitions in non-equilibrium systems: with an application to a non-algorithmic dynamic message buffer. Eur. J. Sci. Res. 54, 59–74 (2011)Google Scholar
  19. 97.
    T.D. Frank, Multistable pattern formation systems: candidates for physical intelligence. Ecol. Psychol. 24, 220–240 (2012)CrossRefGoogle Scholar
  20. 98.
    T.D. Frank, Psycho-thermodynamics of priming, recognition latencies, retrieval-induced forgetting, priming-induced recognition failures and psychopathological perception, in Psychology of Priming ed. by N. Hsu, Z. Schütt, Chap. 9 (Nova Publ., New York, 2012), pp. 175–204Google Scholar
  21. 104.
    T.D. Frank, Secondary bifurcations in a Lotka-Volterra model for n competitors with applications to action selection and compulsive behaviors. Int. J. Bifurcation Chaos 24, article 1450156 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. 105.
    T.D. Frank, Domains of attraction of walking and running attractors are context dependent: illustration for locomotion on tilted floors. Int. J. Sci. World 3, 81–90 (2015)ADSCrossRefGoogle Scholar
  23. 106.
    T.D. Frank, On the interplay between order parameter and system parameter dynamics in human perceptual-cognitive-behavioral systems. Nonlinear Dynamics Psychol. Life Sci. 19, 111–146 (2015)ADSMathSciNetGoogle Scholar
  24. 107.
    T.D. Frank, Formal derivation of Lotka-Volterra-Haken amplitude equations of task-related brain activity in multiple, consecutively preformed tasks. Int. J. Bifurcation Chaos 10, article 1650164 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 109.
    T.D. Frank, Perception adapts via top-down regulation to task repetition: a Lotka-Volterra-Haken modelling analysis of experimental data. J. Integr. Neurosci. 15, 67–79 (2016)CrossRefGoogle Scholar
  26. 110.
    T.D. Frank, A synergetic gait transition model for hysteretic gait transitions from walking to running. J. Biol. Syst. 24, 51–61 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 112.
    T.D. Frank, Determinism of behavior and synergetics, in Encyclopedia of Complexity and Systems Science, ed. by R.A. Meyers, Chap. 695 (Springer, Berlin, 2018)Google Scholar
  28. 114.
    T.D. Frank, J. O’Leary, Motion-induced blindness and the spinning dancer paradigm: a neuronal network approach based on synergetics, in Horizon in Neuroscience Research, ed. by A. Costa, E. Villalba, vol. 35, Chap. 2 (Nova Publ., New York, 2018), pp. 51–80Google Scholar
  29. 120.
    T.D. Frank, M.J. Richardson, S.M. Lopresti-Goodman, M.T. Turvey, Order parameter dynamics of body-scaled hysteresis and mode transitions in grasping behavior. J. Biol. Phys. 35, 127–147 (2009)CrossRefGoogle Scholar
  30. 121.
    T.D. Frank, J. van der Kamp, G.J.P. Savelsbergh, On a multistable dynamic model of behavioral and perceptual infant development. Dev. Psychobiol. 52, 352–371 (2010)CrossRefGoogle Scholar
  31. 125.
    R.W. Frischholz, F.G. Boebel, K.P. Spinner, Face recognition with the synergetic computer, in Proceedings of the First International Conference on Applied Synergetics and Synergetic Engineering (Frauenhofer Institute IIS., Erlangen, 1994), pp. 100–106Google Scholar
  32. 127.
    A. Fuchs, H. Haken, Pattern recognition and associative memory as dynamical processes in a synergetic system. I. Translational invariance, selective attention and decomposition of scene. Biol. Cybern. 60, 17–22 (1988)Google Scholar
  33. 128.
    G. Gambino, M.L. Lombardo, M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion. Math. Comput. Simul. 82, 1112–1132 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 129.
    G. Gambino, M.C. Lombardo, M. Sammartino, V. Sciacca, Turing pattern formation in the Brusselator system with nonlinear diffusion. Phys. Rev. E 88, article 042925 (2013)Google Scholar
  35. 134.
    M.E. Gilpin, Limit cycles in competition communities. Am. Nat. 109, 51–60 (1975)CrossRefGoogle Scholar
  36. 137.
    N.S. Goel, S.C. Maitra, E.W. Montroll, On the Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43, 231–276 (1971)ADSMathSciNetCrossRefGoogle Scholar
  37. 140.
    N.J. Gotelli, A Primer of Ecology (Sinauer Associates, Sunderland, 2008)Google Scholar
  38. 143.
    T.M. Griffin, R. Kram, S.J. Wickler, D.F. Hoyt, Biomechanical and energetic determinants of the walk-trot transition in horses. J. Exp. Biol. 207, 4215–4223 (2004)CrossRefGoogle Scholar
  39. 156.
    H. Haken, Light: Laser Light Dynamics (North Holland, Amsterdam, 1991)Google Scholar
  40. 157.
    H. Haken, Synergetic Computers and Cognition (Springer, Berlin, 1991)zbMATHCrossRefGoogle Scholar
  41. 160.
    H. Haken, Synergetics: Introduction and Advanced Topics (Springer, Berlin, 2004)CrossRefGoogle Scholar
  42. 165.
    M. Hirsch, B. Baird, Computing with dynamic attractors in neural networks. Bio Syst. 34, 173–195 (1995)Google Scholar
  43. 186.
    S. Kim, T.D. Frank, Body-scaled perception is subjected to adaptation when repetitively judging opportunities for grasping. Exp. Brain Res. 234, 2731–2743 (2016)CrossRefGoogle Scholar
  44. 187.
    S. Kim, T.D. Frank, Correlations between hysteretic categorical and continuous judgments of perceptual stimuli supporting a unified dynamical systems approach to perception. Perception 47, 44–66 (2018)CrossRefGoogle Scholar
  45. 215.
    S.M. Lopresti-Goodman, M.T. Turvey, T.D. Frank, Behavioral dynamics of the affordance “graspable”. Atten. Percept. Psychophys. 73, 1948–1965 (2011)Google Scholar
  46. 217.
    A.J. Lotka, The growth of mixed populations. two species competing for a common food supply. J. Wash. Acad. Sci. 23, 461–469 (1932)Google Scholar
  47. 236.
    R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 242.
    R. Müller, D. Cerra, P. Reinartz, Synergetics framework for hyperspectral image classification, in The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 2013 (International Society for Photogrammetry and Remote Sensing, Hannover, 2013), pp. 257–262CrossRefGoogle Scholar
  49. 247.
    A.C. Newell, J.A. Whitehead, Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303 (1969)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. 252.
    G. Nicolis, Introduction to Nonlinear Sciences (Cambridge University Press, Cambridge, 1995)Google Scholar
  51. 259.
    A. Pansuwan, C. Rattanakul, Y. Lenbury, D.J. Wollkind, L. Harrison, L. Rajapakse, K. Cooper, Nonlinear stability analysis of pattern formation on solid surfaces during ion-sputtered erosion. Math. Comput. Model. 41, 939–964 (2005)zbMATHCrossRefGoogle Scholar
  52. 260.
    B. Pena, C. Perez-Garcia, Selection and competition of Turing patterns. Europhys. Lett. 51, 300–306 (2000)ADSCrossRefGoogle Scholar
  53. 265.
    M.I. Rabinovich, M.K. Muezzinoghu, I. Strigo, A. Bystritsky, Dynamic principles of emotion-cognition interaction: mathematical images of mental disorders. PLoS One 5, e12547 (2010)ADSCrossRefGoogle Scholar
  54. 266.
    C. Rattanakul, Y. Lenbury, D.J. Wollkind, V. Chatsudthipong, Weakly nonlinear analysis of a model of signal transduction pathway. Nonlinear Anal. 71, e1620–e1625 (2009)zbMATHCrossRefGoogle Scholar
  55. 282.
    M. Schmutz, W. Banzhaf, Robust competitive networks. Phys. Rev. A 45, 4132–4145 (1992)ADSCrossRefGoogle Scholar
  56. 292.
    L.A. Segal, The nonlinear interaction of two disturbances in thermal convection problem. J. Fluid Mech. 14, 97–114 (1962)ADSMathSciNetCrossRefGoogle Scholar
  57. 293.
    L.A. Segal, The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21, 359–384 (1965)ADSCrossRefGoogle Scholar
  58. 299.
    H. Shimizu, Y. Yamaguchi, Synergetic computer and holonics: information dynamics of a semantic computer. Phys. Scripta 36, 970–985 (1987)ADSCrossRefGoogle Scholar
  59. 308.
    L.E. Stephenson, D.J. Wollkind, Weakly nonlinear analysis of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model system. J. Math. Biol. 33, 771–815 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 314.
    Y. Sumino, K. Yoshikawa, Self-motion of an oil droplet: a simple physiochemical model of active Brownian motion. Chaos 18, 026106 (2008)ADSCrossRefGoogle Scholar
  61. 341.
    W. Wang, Y. Lin, H. Wang, H. Liu, Y. Tan, Pattern selection in an epidemic model with self and cross diffuion. J. Biol. Syst. 19, 19–31 (2011)zbMATHCrossRefGoogle Scholar
  62. 349.
    J. Wesfreid, Y. Pomeau, M. Dubois, C. Normand, P. Berge, Critical effects in Rayleig-Benard convection. Le J. de Phys. Lett. 7, 726–731 (1978)Google Scholar
  63. 354.
    D.J. Wollkind, L.E. Stephenson, Chemical Turing pattern formation analyses: comparison of theory with experiment. SIAM J. Appl. Math. 61, 387–431 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 358.
    A.A. Yudashkin, Bifurcations of steady-state solutions in the synergetic neural network and control of pattern recognition. Auto Remote Control 57, 1647–1653 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Till Frank
    • 1
  1. 1.Dept. Psychology and PhysicsUniversity of ConnecticutStorrsUSA

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